The goal of this section is to show there is a functor \(\mathcal {L}: \mathsf {SMod} \rightarrow \mathsf {SMod}\)  for which behavioural equivalence \(\approx _\mathcal {L}\)  coincides with
weighted language equivalence \(\sim _l\)  of \(K\) -linear weighted automata, extending the
results of Bonchi et al. [1]} to the category of
semimodules. Although the functor \(\mathcal {L}\)  appears to be the same as in
[1]} the underlying details are different. They are
based on a generalization of the notion of a linear weighted automaton
in [3]}, [1]} from the setting of vector
spaces to semimodules. We propose definitions of a \(K\) -linear relation and a
\(K\) -linear bisimulation that generalize the notions of a linear relation and a
linear bisimulation from [3]}. This is the central part of
the paper where the definitions and proofs do not mirror those in
[3]}, [1]}. In particular the notion of
subspace of a vector space is replaced by that of the kernel (in the universal
algebraic sense) of a \(K\) -linear map. But it is remarkable that all proofs go
through and we obtain a true generalization of the concepts from vector spaces
to semimodules.
